Stefano Varricchio

Learning functions represented as multiplicity automata

We study the learnability of multiplicity automata in Angluins exact learning model, and we investigate its applications. Our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. With this theorem in hand, we present a new simple algorithm for learning multiplicity automata with improved time and query complexity, and we prove the learnability of various concept classes. These include (among others): -The class of disjoint DNF, and more generally satisfy-O(1) DNF.-The class of polynomials over finite fields.-The class of bounded-degree polynomials over infinite fields.-The class of XOR of terms.-Certain classes of boxes in high dimensions.In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over GF(2). While multiplicity automata are shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, we study the limitations of this method. We prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulas or even k-term DNF for k = ω(log n) or satisfy-s DNF formulas for s = ω(1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with super-polynomial number of states.

Learning Behaviors of Automata from Multiplicity and Equivalence Queries

We consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field

On the applications of multiplicity automata in learning

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Well quasi-orders and regular languages

of rational numbers (

Words, univalent factors, and boxes

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Finite automata and non-self-embedding grammars

-automaton) from multiplicity and equivalence queries. We provide an algorithm which is polynomial in the size of the

A finiteness condition for semigroups generalizing a theorem of Hotzel

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On the generalization of Higman and Kruskal's theorems to regular languages and rational trees

-automaton and in the maximum length of the given counterexamples. As a consequence, we have that

Finiteness and iteration conditions for semigroups

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COMBINATORIAL PROPERTIES OF UNIFORMLY RECURRENT WORDS AND AN APPLICATION TO SEMIGROUPS

-automata are probably approximately correctly learnable (PAC-learnable) in polynomial time when multiplicity queries are allowed. A corollary of this result is that regular languages are polynomially predictable using membership queries with respect to the representation of unambiguous nondeterministic automata. This is important since there are unambiguous automata such that the equivalent deterministic automaton has an exponentially larger number of states.